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Significance Tests

- THE BASICS
- Could it happen by chance alone?

Statistical Inference

- Confidence IntervalsUse when you want to

estimate a population parameter - Significance TestsUse when you want to assess

the evidence provided by data about some claim

concerning a population - AN OUTCOME THAT WOULD RARELY HAPPEN BY CHANCE IF

A CLAIM WERE TRUE IS GOOD EVIDENCE THAT THE CLAIM

IS NOT TRUE

Overview of a Significance Test

- A test of significance is intended to assess the

evidence provided by data against a null

hypothesis H0 in favor of an alternate hypothesis

Ha. - The statement being tested in a test of

significance is called the null hypothesis.

Usually the null hypothesis is a statement of no

effect or no difference. - A one-sided alternate hypothesis exists when we

are interested only in deviations from the null

hypothesis in one direction - H0 ?0
- Ha ?gt0 (or ?lt0)
- If the problem does not specify the direction of

the difference, the alternate hypothesis is

two-sided - H0 ?0
- Ha ??0

HYPOTHESES

- NOTE Hypotheses ALWAYS refer to a population

parameter, not a sample statistic. - The alternative hypothesis should express the

hopes or suspicions we have BEFORE we see the

data. Dont cheat by looking at the data first.

CONDITIONS

- These should look the same as in the last chapter

(for confidence intervals) - SRS
- Normality
- For meanspopulation distribution is Normal or

you have a large sample size (n30) - For proportions--np10 and n(1-p)10
- Independence

CAUTION

- Be sure to check that the conditions for running

a significance test for the population mean are

satisfied before you perform any calculations.

Test Statistic

- A test statistic comes from sample data and is

used to make decisions in a significance test - Compare sample statistic to hypothesized

parameter - Values far from parameter give evidence against

the null hypothesis (H0) - Standardize your sample statistic to obtain your

TEST STATISTIC

P-values statistical significance

- The probability (computed assuming H0 is true)

that the test statistic would take a value as

extreme or more extreme than that actually

observed is called the P-value of the test. The

smaller the P-value, the stronger the evidence

against the null hypothesis provided by the data. - Significant in the statistical sense doesnt

mean important. It means simply not likely to

happen just by chance. - The significance level a is the decisive value of

the P-value. It makes not likely more exact. - If the P-value is as small or smaller than a, we

say that the data is statistically significant at

level a.

INFERENCE TOOLBOX (p 705)

DO YOU REMEMBER WHAT THE STEPS ARE???

Steps for completing a SIGNIFICANCE TEST

- 1PARAMETERIdentify the population of interest

and the parameter you want to draw a conclusion

about. STATE YOUR HYPOTHESES! - 2CONDITIONSChoose the appropriate inference

procedure. VERIFY conditions (SRS, Normality,

Independence) before using it. - 3CALCULATIONSIf the conditions are met, carry

out the inference procedure. - 4INTERPRETATIONInterpret your results in the

context of the problem. CONCLUSION, CONNECTION,

CONTEXT(meaning that our conclusion about the

parameter connects to our work in part 3 and

includes appropriate context)

Step 1PARAMETER

- Read through the problem and determine what we

hope to show through our test. - Our null hypothesis is that no change has

occurred or that no difference is evident. - Our alternative hypothesis can be either one or

two sided. - Be certain to use appropriate symbols and also

write them out in words.

Step 2CONDITIONS

- Based on the given information, determine which

test should be used. Name the procedure. - State the conditions.
- Verify (through discussion) whether the

conditions have been met. For any assumptions

that seem unsafe to verify as met, explain why. - Remember, if data is given, graph it to help

facilitate this discussion - For each procedure there are several things that

we are assuming are true that allow these

procedures to produce meaningful results.

Step 3CALCULATIONS

- First write out the formula for the test

statistic, report its value, mark the value on

the curve. - Sketch the density curve as clearly as possible

out to three standard deviations on each side. - Mark the null hypothesis and sample statistic

clearly on the curve. - Calculate and report the P-value
- Shade the appropriate region of the curve.
- Report other values of importance (standard

deviation, df, critical value, etc.)

Step 4INTERPRETATION

- There are really two parts to this step

decision conclusion. TWO UNIQUE SENTENCES. - Based on the P-value, make a decision. Will you

reject H0 or fail to reject H0. - If there is a predetermined significance level,

then make reference to this as part of your

decision. If not, interpret the P-value

appropriately. - Now that you have made a decision, state a

conclusion IN THE CONTEXT of the problem. - This does not need to, and probably should not,

have statistical terminology involved. DO NOT

use the word prove in this statement.

Example 1

- Your buddy (Jake) claims to be an A student

(meaning he has a 90 average). You dont know

all of his grades but based on what you have seen

you think this claim is an overstatement. You

took a simple random sample of his grades and

recorded them. They are 92, 87, 86, 90, 80,

91. You also know that all his grades in the

class have a standard deviation of 3.5.

Step 1

- We want to determine whether Jake is accurate in

his measure of his course grade. - Our null hypothesis is that Jake has a course

average of 90. - Our alternative hypothesis is that Jakes course

average is below a 90. - H0 ? 90
- Ha ? lt 90

Step 2

- Since we know the population standard deviation

we will be performing a z-test of significance. - We were told that our selection of grades was an

SRS of Jakes scores. - The box plot shows moderate left skewness. Our

sample is not large so we must assume that the

population of all of Jakes grades are

approximately normal in distribution in order for

our sampling distribution to be approximately

normal. Using the IQR(1.5) method for

determining outliers we see that there are no

outliers in this sample of grades. - Provided Jake has at least 60 overall grades, we

are safe assuming independence and using the

necessary formula for standard deviation.

Step 3

- A curve should be drawn, labeled, and shaded.
- You can use the formula to calculate your z test

statistic for this problem - ? In this case z -1.6330
- Mark this on your sketch.
- Based on our calculations the P-value is 0.0512.
- , s3.5, n6

Step 4

- Since there is no predetermined level of

significance if we are seeking to make a

decision, this could be argued either way. If

Jake were correct about being an A student, we

would only get a sample of grades with an average

this low in roughly 5.1 of all samples. - There is not overwhelming evidence against H0,

however, this is enough to convince me that H0

can be rejected. - Our evidence may not be strong enough to convince

Jake that he is wrong. However, based on this

evidence, I do not believe Jake is accurate about

his average being a 90. It doesnt appear that

Jake is the A student he claims to be.

WARNINGS

- Tests of significance assess evidence against H0
- If the evidence is strong, reject H0 in favor of

Ha - Failure to find evidence against H0 means only

that data are consistent with H0, not that we

have clear evidence that H0 is true - If you are going to make a decision based on

statistical significance, then the significance

level a should be stated before the data are

produced.